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New Stick Number Bounds from Random Sampling of Confined Polygons

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New Stick Number Bounds from Random Sampling of Confined Polygons New Stick Number Bounds from Random Sampling of Confined Polygons Thomas D. Eddy and Clayton Shonkwiler Department of Mathematics, Colorado State University, Fort Collins, CO Abstract The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds on the stick number. We adopt a Monte Carlo approach to finding better bounds, producing very large ensembles of random polygons in tight confinement to look for new examples of knots constructed from few segments. We generated a total of 220 billion random polygons, yielding either the exact stick number or an improved upper bound for more than 40% of the knots with 10 or fewer crossings for which the stick number was not previously known. We summarize the current state of the art in Appendix A, which gives the best known bounds on stick number for all knots up to 10 crossings. 1 Introduction The stick number of a knot is the minimum number of segments needed to create a polygonal version of the knot. The equilateral stick number is defined similarly, though with the added restriction that all the segments should be the same length. Despite their elementary definitions, these invariants are only poorly understood: for example, prior to our work the exact stick number was only known for 30 of the 249 nontrivial knots up to 10 crossings. The sheer simplicity of its definition partially explains the appeal of the stick number as a knot invariant, but the stick number is also interesting as a prototypical geometric or physical knot invariant since the number of segments needed to build a knot is a measure of the geometric complexity of the knot. Polygonal knots serve as a model of ring polymers like bacterial DNA (see the excellent survey [44]) so, while this model is not particularly realistic without adding further constraints, the stick number gives an indication of the minimum size of a ring polymer (or, indeed, any knotted physical system) needed to achieve a given knot type. Any example of a polygonal knot gives an upper bound on the stick number of the knot: if we can build a polygonal version of a knot K out of 10 sticks, then the stick number of K cannot possibly be bigger than 10. Consequently, we can prove state-of-the-art bounds on stick numbers by constructing examples of knots with fewer sticks than previously observed. If we happen to construct an example of a knot K with n sticks where n is known to be a lower bound on the stick number of K, then this proves that the stick number of K is exactly n. 1 935 939 943 945 948 Figure 1: Minimal stick equilateral representations of the knots 935, 939, 943, 945, and 948. Each knot is shown in orthographic perspective, viewed from the direction of the positive z-axis. Our approach to finding such examples is to randomly sample the space of equilateral n-stick knots for small n, which requires identifying the knot type of each sample and checking to see if it beats the previous record. Early work of Millett [36, 37] and Calvo and Millett [13] used a similar Monte Carlo approach, but the challenge is that the overwhelming majority of samples are not interesting, so getting good results requires generating a truly vast number of samples. Indeed, the best extant bounds on (equilateral) stick number are due to Scharein [49] and Rawdon and Scharein [47], who used a combination of stochastic agitation and relaxation starting from standard (rather than random) conformations. The main novelty of our Monte Carlo approach is that we generate polygonal knots with equal-length edges in very tight confinement. Since complicated knots are typically condensed, this approach is a form of enriched sampling which boosts the yield of the complicated knots we seek. Generating confined knots to explore stick number is not new in itself – Calvo and Millett [13, 37] generated stick knots with vertices uniformly distributed in the unit cube (the so-called uniform random polygon model) – but ours seems to be the first such investigation which addresses equilateral stick number. Our strategy depends, in turn, on recent advances in sampling random equilateral polygons using symplectic geometry [16, 15]. The efficiency of our algorithms combined with modern hardware leads to a substantial improvement 2 over earlier Monte Carlo experiments in the sheer size of the ensembles we produce. We generated a total of 220 billion random knots, requiring approximately 50,000 core-hours of processing time on a 64-core machine with 2.3 GHz AMD Opteron 6276 processors. The bulk of this time was devoted to identifying the knot type of each of these 220 billion knots, which we did using the HOMFLY polynomial, hyperbolic volume, and crossing number. 1 We found examples of the knots 935, 939, 943, 945, and 948 built with 9 equal-length sticks, shown in Figure 1; since the stick number of these knots is known to be at least 9, this gives the exact (equilateral) stick number for these five knots: Theorem 1. The stick number and equilateral stick number of each of the knots 935, 939, 943, 945, and 948 is exactly 9. As we will see, this implies that each of these knots has superbridge index 4. Also, we found new upper bounds on the equilateral stick number for 88 of the 214 knots up to 10 crossings for which it remains unknown: Theorem 2. The equilateral stick number of each of the knots 92, 93, 911, 915, 921, 925, 927, 108, 1016, 1017, 1056, 1083, 1085, 1090, 1091, 1094, 10103, 10105, 10106, 10107, 10110, 10111, 10112, 10115, 10117, 10118, 10119, 10126, 10131, 10133, 10137, 10138, 10142, 10143, 10147, 10148, 10149, 10153, and 10164 is less than or equal to 10. The equilateral stick number of each of the knots 103, 106, 107, 1010, 1015, 1018, 1020, 1021, 1022, 1023, 1024, 1026, 1028, 1030, 1031, 1034, 1035, 1038, 1039, 1043, 1044, 1046, 1047, 1050, 1051, 1053, 1054, 1055, 1057, 1062, 1064, 1065, 1068, 1070, 1071, 1072, 1073, 1074, 1075, 1077, 1078, 1082, 1084, 1095, 1097, 10100, and 10101 is less than or equal to 11. The equilateral stick number of each of the knots 1076 and 1080 is less than or equal to 12. In particular, all knots up to 10 crossings have equilateral stick number ≤ 12. An upper bound on equilateral stick number implies an upper bound on stick number, and all of the corresponding stick number bounds in Theorem 2 are new except for 10107, 10119, and 10147. These knots, together with 819, 929, 1016, and 1079, appeared on Rawdon and Scharein’s list of knots for which they could not find an equilateral stick knot achieving a known bound on stick number, and thus were potential examples of knots for which stick number and equilateral stick number differ. Millett [38] found an example of an equilateral 8-stick 819, proving that eqstick(819) = 8, which was already known to be the stick number, and our 10-stick 1016 (see Figure 2) beats the previous best bounds on both stick number (11) and equilateral stick number (12). In turn, our observed equilateral 10-stick examples of 10107, 10119, and 10147 match the best previous bound on stick number, leaving only 929 and 1079 from Rawdon and Scharein’s list. The new bounds on equilateral stick number for the knots 108, 1016, 1039, 1064, 1073, 10105, 10110, and 10117 in Theorem 2 are 2 smaller than the previous best bounds due to Rawdon and Scharein, and for 1084 (see Figure 2) the new bound is 3 smaller. These substantial improvements in bounds for certain knots, coupled with the fact that Rawdon and Scharein’s bounds are superior to ours for the knots 916, 101, 10109, 10113, 10114, and 10123, gives a sense of the differences between their approach and ours, which we view as complementary. 1For knots up to 10 crossings we use Alexander–Briggs notation consistent with the (Perko-corrected) Rolfsen table [48]; for knots with 11 or more crossings, we use the Dowker–Thistlethwaite naming convention. 3 1016 1084 Figure 2: The 10-stick 1016 improves the best previous bound on equilateral stick number by 2. The 11-stick 1084 improves the best previous bounds on both stick number and equilateral stick number by 3. Each knot is shown in orthographic perspective from the direction (3;1;1). Organization In building up to Theorems1 and2, we start with background on stick number and equilateral stick number in Section 2 and a review of new approaches to sampling random polygons using symplectic geometry in Section 3. The latter section may be of independent interest, since it gives a concise and reasonably self-contained exposition of the symplectic approach to sampling confined polygons introduced in [16]. We describe our computational pipeline in Section 4 and give a more detailed statement of our results in Section 5, which is supplemented by the tables in the appendices. Finally, we pose some questions and mention some directions for future inquiry in Section 6. This is work that grew out of the first author’s master’s thesis [22], which was based on smaller ensembles of random knots. 2 Stick Number A tame, piecewise-linear embedding S1 ,! R3 is called a polygonal knot or a stick knot. Any stick knot consists of a finite number of straight segments. Define the stick number of a knot K, denoted stick(K), to be the minimum number of linear segments necessary to construct a polygonal version of K.
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